A Dynamic Programming Model of Fractional Flows with Application to Maintenance and Replacement Problems

This paper studies a deterministic dynamic programming model of fractional flows in an n-sector system, characterizing its solutions under general conditions, and developing algorithms for their computation. The calculations that the latter entail are not much more difficult than those needed for an n-state Markovian decision process (in certain restrictive cases, the model is “isomorphic” to an n-state MDP). Some of our results extend to a stochastic model as a “certainty equivalence.” Applications include a maintenance/replacement/sales model, a manpower retention model that extends Flynn (Flynn, J. 1970. Linear Production Models with Incentives: A Dynamic Programming Approach. Ph.D. dissertation, University of California, Berkeley; Flynn, J. 1975. Retaining productive units: A dynamic programming model with a good steady state solution. Management Sci.21 753–764.), Grinold-Stanford (Grinold, R., R. Stanford. 1974. Optimal control of a graded manpower system. Management Sci.8 1201–1216.) and Stanford (Stanford, R. 1976. Analytical solution of a dynamic transaction. Math. Programming10 214–229.), and a constrained version of Pliska’s (Pliska, S. 1976. Optimization of multitype branching processes. Management Sci.23 117–124.) branching model.